A Probabilistic Study of Bound Consistency for the Alldifferent Constraint

This paper introduces a mathematical model for bound consistency of the con- straint AllDifferent. It allows us to compute the probability that the filtering algorithm effectively removes at least one value in the variable domains. A complete study of the bound consistency properties is then proposed. It identifies several behaviors depending on some macroscopic quantities related to the variables and the domains. Finally, it is shown that the probability for an AllDifferent constraint to be bound consistent can be asymptotically estimated in constant time. The experiments illustrate that the precision is good enough for a practical use in constraint programming.